Budgies can generally be kept in a flock, as long as there is no breeding – meaning no nests, no nest boxes, bird huts or tents or anything that can be used as a nest, including large food dishes. You need to have as many food and water cups as birds, plus one extra, at different places in the cage. This prevents one bird from keeping others away from the food and water. You also should only have an even number of birds. When there is an odd number, all of the birds tend to pair off, regardless of sex, and the odd bird out gets bullied.
However, I would recommend finding homes for the three offspring of the one pair. Parent birds will rarely tolerate having grown offspring living with them. It’s against their instinct, because related birds should never breed. And the 3 offspring should be separated by sex, again because related birds should not be allowed to breed with each other. If all are together, and if the parents do tolerate the offspring, at some point the brothers and sisters will mate or the parents will mate with their offspring. Even if you don’t plan to let them sit on eggs, things happen, you might decide you can’t throw the eggs away, but then you end up with chicks with genetic issues, if they survive. So the best thing to do is find them a new home. You can’t just keep the three together – even if all three are male or female, one bird will eventually be the odd bird and be bullied.
All of that being said, budgies are extremely territorial, especially when breeding. And since the one pair has now reproduced, they might remain territorial. The two pairs might get along in one big cage, but they also might not. So keep this in mind when putting them together. Watch them closely for the first couple of weeks and be ready to separate them if there are any signs of fighting.
Thank you for asking Lafeber,